Newspace parameters
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.07611045255\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.564.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 5x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 5x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.67282 | 0 | 1.00000 | 0 | −1.14399 | 0 | 4.14399 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.35194 | 0 | 1.00000 | 0 | 4.17226 | 0 | −1.17226 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.32088 | 0 | 1.00000 | 0 | −5.02827 | 0 | 8.02827 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 260.2.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2340.2.a.n | 3 | ||
| 4.b | odd | 2 | 1 | 1040.2.a.o | 3 | ||
| 5.b | even | 2 | 1 | 1300.2.a.i | 3 | ||
| 5.c | odd | 4 | 2 | 1300.2.c.f | 6 | ||
| 8.b | even | 2 | 1 | 4160.2.a.bo | 3 | ||
| 8.d | odd | 2 | 1 | 4160.2.a.br | 3 | ||
| 12.b | even | 2 | 1 | 9360.2.a.da | 3 | ||
| 13.b | even | 2 | 1 | 3380.2.a.o | 3 | ||
| 13.d | odd | 4 | 2 | 3380.2.f.h | 6 | ||
| 20.d | odd | 2 | 1 | 5200.2.a.ci | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.2.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1040.2.a.o | 3 | 4.b | odd | 2 | 1 | ||
| 1300.2.a.i | 3 | 5.b | even | 2 | 1 | ||
| 1300.2.c.f | 6 | 5.c | odd | 4 | 2 | ||
| 2340.2.a.n | 3 | 3.b | odd | 2 | 1 | ||
| 3380.2.a.o | 3 | 13.b | even | 2 | 1 | ||
| 3380.2.f.h | 6 | 13.d | odd | 4 | 2 | ||
| 4160.2.a.bo | 3 | 8.b | even | 2 | 1 | ||
| 4160.2.a.br | 3 | 8.d | odd | 2 | 1 | ||
| 5200.2.a.ci | 3 | 20.d | odd | 2 | 1 | ||
| 9360.2.a.da | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 2T_{3}^{2} - 8T_{3} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(260))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} - 2 T^{2} - 8 T + 12 \)
$5$
\( (T - 1)^{3} \)
$7$
\( T^{3} + 2 T^{2} - 20 T - 24 \)
$11$
\( T^{3} - 24T + 36 \)
$13$
\( (T - 1)^{3} \)
$17$
\( T^{3} - 2 T^{2} - 36 T - 24 \)
$19$
\( T^{3} - 8 T^{2} - 16 T + 164 \)
$23$
\( T^{3} + 10 T^{2} + 24 T + 12 \)
$29$
\( T^{3} - 10 T^{2} + 12 T + 24 \)
$31$
\( T^{3} + 12 T^{2} + 24 T + 4 \)
$37$
\( T^{3} + 2 T^{2} - 44 T - 72 \)
$41$
\( T^{3} + 2 T^{2} - 36 T + 24 \)
$43$
\( T^{3} + 2 T^{2} - 8 T - 12 \)
$47$
\( T^{3} + 10 T^{2} + 12 T - 24 \)
$53$
\( T^{3} + 18 T^{2} + 12 T - 648 \)
$59$
\( T^{3} + 16T^{2} - 564 \)
$61$
\( T^{3} - 14 T^{2} + 44 T + 8 \)
$67$
\( T^{3} - 14 T^{2} + 20 T + 152 \)
$71$
\( T^{3} - 24T + 36 \)
$73$
\( T^{3} - 14 T^{2} - 124 T + 1784 \)
$79$
\( T^{3} - 8 T^{2} - 16 T + 32 \)
$83$
\( T^{3} + 6 T^{2} - 132 T - 936 \)
$89$
\( T^{3} + 2 T^{2} - 180 T + 216 \)
$97$
\( T^{3} - 26 T^{2} + 140 T + 8 \)
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